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## C. Calculation of the stress tensor

We now must calculate under the conditions given in Eqs. (6.50), (6.51) and (6.52). Let us write the flow like . It is not difficult to understand that under these conditions, the equations of motion read

 = (6.58) = (6.59) = (6.60) = (6.61)

Each bead simply gets an extra velocity equal to the average velocity at its instantaneous position.

The reader who wishes to explicitly check the above equations, may start with Eq. (4.2) and replace the particles velocity in the friction force by its velocity relative to the average velocity, i.e. replace by . Going all the way to Eq. (4.39) he/she will find that also there is replaced by . Next, putting equal to zero, and solving for , he/she will find an extra term . This finally will lead to an extra term in the Smoluchowski equation, which can only be obtained from the Langevin equation  Eq. (4.27) if it is augmented with a term on the right hand side.

We now continue our calculation of the stress tensor. To this end, we transform to normal coordinates

 = (6.62) = (6.63)

Using these equations of motion for the shear rate defined by Eqs. (6.50) to (6.52) we derive

 = = (6.64)

At the end of this section we shall argue that the last two terms vanish. Moreover, for small values of we shall approximate by its equilibrium value . Using , we find from Eq. ( 6.64)

 = (6.65) = (6.66)

Combining Eqs. (6.49), (6.57) and (6.65) we find an integral expression for the stress tensor :

 (6.67)

In polymer melts, we must neglect the solvent contribution . We recognize that the contribution of the Rouse chains to the shear relaxation modulus  is given by

 (6.68)

So the viscosity  of a Rouse melt, at constant monomer concentration c, is proportional to N:
 = (6.69)

This has been confirmed for polymer melts with low molecular weight. Polymer melts of high molecular weight give different results, stressing the importance of so-called entanglements . We will deal with this in chapter 8.

In dilute solutions, we do not neglect the solvent contribution . We combine Eqs. (6.53), (6.57) and ( 6.66) to obtain an expression for the intrinsic viscosity ,

 = = (6.70)

Here, is the polymer concentration; M is the mol mass of the polymer, and NAv is Avogadro's number.

We finish this section by calculating . Integrating Eq. (6.62) we find

 (6.71)

Because of causality for . Then
 = = = (6.72)

i.e. .

Next: Summary Up: Viscosity of a dilute Previous: B. The stress tensor
W.J. Briels